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Commutativity for a certain class of rings. (English) Zbl 0839.16029

For rings \(R\), the author considers four similar conditions involving commutators, of which the first is \((c_1)\): “For every \(x, y\in R\) there holds \(x^r[x^s, y]= \pm[ x, y^t] x^n\) with integers \(t> 1\), \(s\geq 1\), \(n\geq 0\), \(r\geq 0\).” His theorems assert that a ring \(R\) with 1 must be commutative if it satisfies one of these conditions in conjunction with an appropriate restriction on additive torsion. Apparently \(t\), \(s\), \(n\), and \(r\) are fixed, but it is not clear how to interpret the ambiguity in sign in condition \((c_1)\) and the other three conditions. The safe interpretation is that one chooses the same sign for all \(x, y\in R\).
While the theorems of this paper are similar to many known results, the proofs do have an element of novelty; they provide an interesting application of some earlier results of T. P. Kezlan [Math. Jap. 29, 135-139 (1984; Zbl 0538.16028)].

MSC:

16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)

Citations:

Zbl 0538.16028
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References:

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